Chapter 4
Today is
date()## [1] "Tue Nov 17 20:14:20 2020"Analysis exercise
1. Create new Rmarkdown file
done!
2. Load the Boston Data
Let us load the data and then summarize and provide a glimpse into the data.
library(dplyr); library (MASS)
glimpse(Boston) ## Rows: 506
## Columns: 14
## $ crim <dbl> 0.00632, 0.02731, 0.02729, 0.03237, 0.06905, 0.02985, 0.08829…
## $ zn <dbl> 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.5, 12.5, 12.5, 12.5, 12.5, …
## $ indus <dbl> 2.31, 7.07, 7.07, 2.18, 2.18, 2.18, 7.87, 7.87, 7.87, 7.87, 7…
## $ chas <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
## $ nox <dbl> 0.538, 0.469, 0.469, 0.458, 0.458, 0.458, 0.524, 0.524, 0.524…
## $ rm <dbl> 6.575, 6.421, 7.185, 6.998, 7.147, 6.430, 6.012, 6.172, 5.631…
## $ age <dbl> 65.2, 78.9, 61.1, 45.8, 54.2, 58.7, 66.6, 96.1, 100.0, 85.9, …
## $ dis <dbl> 4.0900, 4.9671, 4.9671, 6.0622, 6.0622, 6.0622, 5.5605, 5.950…
## $ rad <int> 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4…
## $ tax <dbl> 296, 242, 242, 222, 222, 222, 311, 311, 311, 311, 311, 311, 3…
## $ ptratio <dbl> 15.3, 17.8, 17.8, 18.7, 18.7, 18.7, 15.2, 15.2, 15.2, 15.2, 1…
## $ black <dbl> 396.90, 396.90, 392.83, 394.63, 396.90, 394.12, 395.60, 396.9…
## $ lstat <dbl> 4.98, 9.14, 4.03, 2.94, 5.33, 5.21, 12.43, 19.15, 29.93, 17.1…
## $ medv <dbl> 24.0, 21.6, 34.7, 33.4, 36.2, 28.7, 22.9, 27.1, 16.5, 18.9, 1…summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00The data contain 506 observations, which each apparently represent different census tracts in Boston. The data have 14 variables that describe various characteristics of these census tracts like crime rate, tax rate, and a median value of homes in the census tract.
3. Graphical overview
I start by drawing some scatter plots.
library(tidyr); library(dplyr); library(ggplot2); library(GGally)
gather(Boston) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar()
3.1 Histograms
Since the data contain so many variables, most of the histograms are relatively small. However, we can at least say that there seems to be quite a bit of variation in age, nox and medv variables. ### 3.1 Scatter and correlations plots
library("ggplot2"); library("GGally");library(corrplot)
pairs(Boston)
cMatrix = cor(Boston) %>% round(digits=2)
corrplot(cMatrix, method = "ellipse", type= "upper")
Since the correlation plot with scatter plots is messy, I will concentrate just on the latter figure, which shows only the correlations between variables. Based on the figure, some variables like lower statuts of the population (lstat) and median value of homes (medv) are very negatively correlated, some variables like proportion of non-retail business (indus) and emissons (nox) are very positively correlated, whereas between some variables like there do not exist any relationship.
4. Standardize the data
4.1 Standardization
library("ggplot2"); library("GGally");library(corrplot)
bostonStand <-scale(Boston)
bostonStand <- as.data.frame(bostonStand)
summary(bostonStand)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865We can see that after the standardization procedure, each variable is centered around zero. (mean is zero). Let us then create a new categorical variable using the existing crime variable.
library("ggplot2"); library("GGally");library(corrplot); library(dplyr)
bins <- quantile(bostonStand$crim)
crime <-cut(bostonStand$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high") )
bostonStand <- dplyr::select(bostonStand , -crim)
summary(bostonStand)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv
## Min. :-1.9063
## 1st Qu.:-0.5989
## Median :-0.1449
## Mean : 0.0000
## 3rd Qu.: 0.2683
## Max. : 2.9865bostonStand$crime <- crime
summary(bostonStand)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.98654.2 Train and test sets
Let us create training and test sets and then have a glimpse at both of them.
library("ggplot2");library(corrplot); library(dplyr)
nObs <- nrow(bostonStand)
indTrain <- sample(nObs , size = nObs * 0.8)
# Training set
trainBoston <- bostonStand[indTrain,]
# Test set
testBoston <- bostonStand[-indTrain,]
glimpse(trainBoston) ## Rows: 404
## Columns: 14
## $ zn <dbl> 0.4560568, 3.3717021, 1.4422310, 2.7285450, -0.4872402, -0.48…
## $ indus <dbl> -0.76917014, -1.19043127, -1.12192167, -1.19334657, -0.079701…
## $ chas <dbl> -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0.2723291, -…
## $ nox <dbl> -1.06746238, -1.33498587, -1.01568364, -1.09335175, -0.566934…
## $ rm <dbl> 0.21116139, 1.14339028, 0.64667596, 0.44172792, 0.12861288, -…
## $ age <dbl> -0.6918540, -1.6972232, -1.3419691, -1.6616978, -1.2886809, 0…
## $ dis <dbl> 1.914535748, 1.667968097, 1.274989030, 0.762715291, 0.0714045…
## $ rad <dbl> -0.2927910, -0.9818712, -0.5224844, -0.7521778, -0.6373311, -…
## $ tax <dbl> -0.46421320, -0.73121670, -0.06074124, -0.92701927, -0.778683…
## $ ptratio <dbl> 0.29768250, -1.45755797, -1.50374851, -0.07184181, 0.06672981…
## $ black <dbl> 0.19755732, 0.41673722, 0.36186010, 0.42670493, 0.31914137, 0…
## $ lstat <dbl> -0.438739148, -0.672597937, -1.115109179, -1.166922204, -0.45…
## $ medv <dbl> 0.21389273, 1.05111278, 0.86627199, 0.89889095, 0.60532029, -…
## $ crime <fct> med_low, low, low, low, med_low, low, med_high, med_high, med…glimpse(testBoston) ## Rows: 102
## Columns: 14
## $ zn <dbl> -0.48724019, 0.04872402, -0.48724019, -0.48724019, -0.4872401…
## $ indus <dbl> -1.3055857, -0.4761823, -0.4368257, -0.4368257, -0.4368257, -…
## $ chas <dbl> -0.2723291, -0.2723291, -0.2723291, -0.2723291, -0.2723291, -…
## $ nox <dbl> -0.83445805, -0.26489191, -0.14407485, -0.14407485, -0.144074…
## $ rm <dbl> 1.227362043, -0.930285282, -0.641365488, -1.017103545, -0.454…
## $ age <dbl> -0.51067434, 1.11638970, -0.42896588, 1.04889141, 0.73271521,…
## $ dis <dbl> 1.076671135, 1.086121629, 0.334118787, 0.001356935, 0.1031753…
## $ rad <dbl> -0.7521778, -0.5224844, -0.6373311, -0.6373311, -0.6373311, -…
## $ tax <dbl> -1.10502160, -0.57694801, -0.60068166, -0.60068166, -0.600681…
## $ ptratio <dbl> 0.11292035, -1.50374851, 1.17530274, 1.17530274, 1.17530274, …
## $ black <dbl> 0.4406159, 0.3281233, 0.4265954, 0.2179309, 0.3927490, 0.4147…
## $ lstat <dbl> -1.02548665, 2.41937935, -0.58577611, 1.17166569, 0.16481258,…
## $ medv <dbl> 1.48603229, -0.65594629, -0.28626471, -0.97126294, -0.3188836…
## $ crime <fct> low, med_low, med_high, med_high, med_high, med_high, med_hig…5. Discriminant analysis
library("ggplot2");library(corrplot); library(dplyr)
lda.fit <- lda( crime ~ . , data = trainBoston )
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(trainBoston$crime)
plot(lda.fit , dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
6. Predict and cross tabulate
Let us first save the initial values of the crime variable of test data into a separate vector.
library("ggplot2");library(corrplot); library(dplyr)
# save the initial values of crime vectors
orginalCrime <- testBoston$crime
testBoston <- dplyr::select(testBoston, -crime)Then let us predict the values of the crime variable in the test data using the model we found using the training data. Finally, we compare the predicted and actual values of the crime variable.
library("ggplot2");library(corrplot); library(dplyr)
lda.pred <- predict(lda.fit, newdata = testBoston)
table(correct = orginalCrime, lda.pred$class )##
## correct low med_low med_high high
## low 11 8 3 0
## med_low 5 16 8 0
## med_high 1 6 15 0
## high 0 0 0 29The diagonal line from the left top corner to the bottom right corner shows the number of correct predictions. Based on the table, our model performs ok. The model does very well at predicting extreme values (low and high crimes) while there seem to be more wrong guesses when predicting values are closer to the median.
7. Distances and K-means algorithm
7.1 Distances
First we re-load the data and standardize the variables. Then I calculate euclidean distances between variables.
library(dplyr); library (MASS)
boston<-scale(Boston)
boston <- as.data.frame(boston)
distE <- dist(boston)
summary(distE )## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.39707.2 K-means algorithm
I first perform k-mean clustering with 7 clusters. Note that I pick the number of clusters just randomly.
library(dplyr); library (MASS)
kM <- kmeans(Boston, centers = 7)
pairs(boston[1:5], col = kM$cluster )
pairs(boston[6:10], col = kM$cluster )
pairs(boston[11:14], col = kM$cluster )
Let us then be a bit formal and find the optimal number of clusters by calculating the total of within-cluster sum of squares (WCSS).
library(dplyr); library (MASS); library("ggplot2");
set.seed(123)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(boston, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')
Based on the figure, it seems that the WCSS drops heavily when the number of clusters is 2. Let us perform the k-mean clustering using two clusters and plot the results
library(dplyr); library (MASS)
kM <- kmeans(Boston, centers = 2)
pairs(boston[1:5], col = kM$cluster )
pairs(boston[6:10], col = kM$cluster )
pairs(boston[11:14], col = kM$cluster )